Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs.

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Title page
Chapter 1. Introduction
Chapter 2. Main Results
2.1. Set-up
2.2. Structural decomposition
2.3. Exponential Trichotomy
2.4. Index Theorems and spectral properties
2.5. Structural stability/instability
2.6. A theorem where does not have a positive lower bound on ₊
2.7. Some Applications to PDEs
Chapter 3. Basic properties of Linear Hamiltonian systems
Chapter 4. Finite dimensional Hamiltonian systems
Chapter 5. Invariant subspaces
5.1. Maximal non-positive invariant subspaces (Pontryagin invariant subspaces)
5.2. Further discussions on invariant subspaces and invariant decompositions
Chapter 6. Structural decomposition
Chapter 7. Exponential trichotomy
Chapter 8. The index theorems and the structure of _{ }
8.1. Proof of Theorem 2.3: the index counting formula
8.2. Structures of subspaces _{ } of generalized eigenvectors
8.3. Subspace of generalized eigenvectors ₀ and index ₀^{≤0}
8.4. Non-degeneracy of ⟨ ⋅,⋅⟩ on _{ } and isolated purely imaginary spectral points
Chapter 9. Perturbations
9.1. Persistent exponential trichotomy and stability: Theorem 2.4 and Proposition 2.9
9.2. Perturbations of purely imaginary spectrum and bifurcation to unstable eigenvalues
Chapter 10. Proof of Theorem 2.7 where (H2.b) is weakened
Chapter 11. Hamiltonian PDE models
11.1. Stability of Solitary waves of Long wave models
11.2. Stability of periodic traveling waves
11.3. Modulational Instability of periodic traveling waves
11.4. The spectral problem = ′
11.5. Stability of steady flows of 2D Euler equation
11.6. Stability of traveling waves of 2-dim nonlinear Schrödinger equations with nonzero conditions at infinity
Appendix A. Appendix
Bibliography
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